1. Fuzzy graph and relation

2. Outline Graph and fuzzy graph Characteristics of fuzzy relations Types of fuzzy relations

3. Graph and fuzzy graph Graph

4. Graph and fuzzy graph Fuzzy graph ~ V : is fuzzy node ~

5. Graph and fuzzy graph Fuzzy graph and fuzzy relation

6. Graph and fuzzy graph α-cut of fuzzy graph

7. Graph and fuzzy graph α-cut of fuzzy graph

8. Graph and fuzzy graph Fuzzy network

9. Characteristics of fuzzy relations Reflexive relation

10. Characteristics of fuzzy relations Symmetric relation ( ( ( ) ) ) ()

11. Characteristics of fuzzy relations An example of perfect antisymmetric

12. Characteristics of fuzzy relations Transitive relation

13. Characteristics of fuzzy relations An example of transitive relation

14. Characteristics of fuzzy relations Transitive closure

15. Characteristics of fuzzy relations An example of transitive closure

16. Types of fuzzy relation Fuzzy equivalence relation Reflexive relation Symmetric relation Transitive relation

17. Types of fuzzy relation abcd a1.00.80.40.1 b0.81.00.0 c0.40.01.00.5 d0.10.00.51.0 (i) Reflexive relation √ (ii) Symmetric relation √ (iii) Transitive relation abcd a1.00.80.4 b0.81.00.40.1 c0.4 1.00.5 d0.40.10.51.0 X

18. Types of fuzzy relation Fuzzy equivalence relation applications – Partition of sets

19. Types of fuzzy relation Fuzzy equivalence relation applications – Partition by α-cut

20. Types of fuzzy relation Fuzzy equivalence relation applications – Set similar to element x Set similar to d Set similar to e

21. Types of fuzzy relation Fuzzy compatibility relation Reflexive relation Symmetric relation

22. Types of fuzzy relation Fuzzy pre-order relation Reflexive relation Transitive relation abc a1.0 0.4 b0.01.00.3 c0.01.0 abc a 0.4 b01.00.3 c01.0

23. Types of fuzzy relation Fuzzy semi-pre-order or nonreflexive fuzzy pre-order relation XReflexive relation Transitive relation abc a0.20.60.3 b0.00.60.3 c0.00.60.3

24. Types of fuzzy relation Anti-reflexive fuzzy pre-order relation Anti-reflexive relation Transitive relation abc a0.00.40.3 b0.0 c

25. Types of fuzzy relation Fuzzy order relation Reflexive relation Antisymmetric relation Transitive relation Perfect fuzzy order

26. Types of fuzzy relation Fuzzy order relation  crisp order

27. Types of fuzzy relation Fuzzy order relation  Dominating class  Dominated class

28. Types of fuzzy relation

29. Dissimilitude relation Antireflexive relation Symmetric relation Min-Max transitive relation

30. Types of fuzzy relation ex Min number = 0.2

31. Types of fuzzy relation ex abcd a0.00.40.10.0 b0.40.00.21.0 c0.10.20.00.3 d0.01.00.30.0 (d,c) ;0.0 v 0.1 = 0.1 1.0 v 0.2 = 1.0 0.3 v 0.0 = 0.3 0.0 v 0.3 = 0.3 Min number = 0.1 (b,c);0.4 v 0.1 = 0.4 0.0 v 0.2 = 0.2 0.2 v 0.0 = 0.2 1.0 v 0.3 = 1.0 Min number = 0.2

32. Types of fuzzy relation Fuzzy morphism  Homomorphism  Fuzzy homomorphism

33. Types of fuzzy relation Fuzzy morphism  Strong homomorphism  Fuzzy strong homomorphism

34. Types of fuzzy relation Fuzzy homomorphism

35. Types of fuzzy relation Fuzzy strong homomorphism